Combinatorics Tools
Browse all available combinatorics tools.
Available tools
- Ballot Prefix Surplus Accumulate explicitly supplied paired vote-block counts into signed prefix surpluses without counting ballot arrangements.
- Binomial Coefficient Log Ratio Step Tool Compute binomial coefficient log ratio step from explicit bounded inputs.
- Binomial Coefficient Symmetry Index Return n-k and min(k,n-k) for the explicit binomial symmetry identity without evaluating the coefficient.
- Binomial Lattice Step Forward Ratio Tool Calculate binomial lattice step forward ratio from explicit caller-supplied inputs.
- Binomial Pascal Single Recurrence Residual Tool Compute binomial pascal single recurrence residual from explicit bounded inputs.
- Bounded Composition Cap Slack Tool Calculate bounded composition cap slack from explicit caller-supplied inputs.
- Catalan Convolution Single Summand Tool Compute catalan convolution single summand from explicit bounded inputs.
- Catalan Number Calculator Compute the nth Catalan number using the multiplicative formula.
- Catalan Number Practice Create reproducible practice for Catalan Number Calculator from supplied scenarios, returning validated questions and computed answers under the source tool's assumptions and limits.
- Catalan Triangulation Diagonal Budget Tool Calculate catalan triangulation diagonal budget from explicit caller-supplied inputs.
- Circular Arrangement Reflection Orbit Size Tool Calculate circular arrangement reflection orbit size from explicit caller-supplied inputs.
- Circular Permutation Anchor Factor Tool Compute circular permutation anchor factor from explicit bounded inputs.
- Combination Colexicographic Rank Exact zero-based colexicographic combination ranking on bounded sorted distinct indexes; results are integer text or explicit index witnesses.
- Combination Colexicographic Unrank Exact zero-based colexicographic combination unranking on bounded sorted distinct indexes; results are integer text or explicit index witnesses.
- Combination Count Calculator Calculate the number of unordered ways to choose k items from n items.
- Combination Lexicographic Rank Exact zero-based lexicographic combination ranking on bounded sorted distinct indexes; results are integer text or explicit index witnesses.
- Combination Lexicographic Unrank Exact zero-based lexicographic combination unranking on bounded sorted distinct indexes; results are integer text or explicit index witnesses.
- Combination Prefix Completion Count Count exact sorted-combination completions of an explicit prefix, excluding earlier skipped indexes; does not enumerate or assume the prefix is arbitrary subset membership.
- Combination Unranking One Binomial Block Tool Compute combination unranking one binomial block from explicit bounded inputs.
- Combinatorics: Binomial Adjacent Coefficient Ratio Tool Compute C(n,k+1)/C(n,k) for one valid k<n without building a binomial row.
- Combinatorics: Combination Complement Index Map Tool Map a sorted unique combination of zero-based indices to its ordered universe complement, not a generic value-set lookup.
- Combinatorics: Combination Prefix Minimum Tail Tool Construct the lexicographically smallest remaining tail of a sorted zero-based combination prefix, with infeasible tails empty and no full combination enumeration.
- Combinatorics: Composition Positive Part Shift Tool Map one ordered positive composition to its weak composition by subtracting one per part, retaining both totals; no partition/count enumeration.
- Combinatorics: Derangement Inclusion Term Tool Return one exact signed n!/k! contribution to the derangement inclusion-exclusion sum, not the entire subfactorial.
- Combinatorics: Dyck Word Up Down Balance Tool Check one bounded up/down word for Dyck prefix nonnegativity and final balance, retaining a negative prefix witness rather than merely equal counts.
- Combinatorics: Integer Partition Durfee Size Tool Compute the Durfee-square side of one descending integer partition, not a partition-number solver.
- Combinatorics: Lattice Path Endpoint Parity Check Tool Check necessary and sufficient reachability for unrestricted four-neighbor square-lattice walks of a supplied length, not obstacle-aware pathfinding.
- Combinatorics: Multiset Multiplicity Factorial Denominator Tool Compute the exact product of multiplicity factorials for a multiset-count denominator; no distinct permutation numerator assumed.
- Combinatorics: Necklace Rotation Stabilizer Quotient Tool Apply orbit-stabilizer to a caller-established rotation action with an exactly dividing stabilizer size; does not enumerate necklaces.
- Combinatorics: Partition Ferrers Cell Count Tool Count cells and extents of a supplied descending Ferrers partition diagram, preserving the empty diagram.
- Combinatorics: Permutation Cycle Parity From Count Tool Derive permutation parity from a caller-established size and disjoint-cycle count including fixed points; no cycle discovery.
- Combinatorics: Restricted Growth String Next Symbol Bound Tool Validate one zero-based restricted-growth-string prefix and return its next-symbol bound, with an empty prefix requiring first symbol zero.
- Combinatorics: Rook Placement Cell Conflict Tool Check whether one candidate rook attacks an explicitly supplied paired rook placement by row or column, without solving a board-packing problem.
- Composition Fixedpart Remainder Budget Tool Compute composition fixedpart remainder budget from explicit bounded inputs.
- Composition Part Upper Slack Tool Calculate composition part upper slack from explicit caller-supplied inputs.
- Composition Zero Part Capacity Tool Inspect a supplied weak composition’s part-count witness against an explicit zero-part allowance, without counting compositions.
- Cyclic Composition Part Count Tool Count positive-part compositions of a total into a fixed number of parts modulo cyclic rotation using Burnside’s lemma, including periodic compositions.
- Derangement Recurrence Single Branch Tool Calculate derangement recurrence single branch from explicit caller-supplied inputs.
- Double Factorial Calculator Calculate the double factorial of a nonnegative integer.
- Double Factorial Practice Create reproducible practice for Double Factorial Calculator from supplied scenarios, returning validated questions and computed answers under the source tool's assumptions and limits.
- Dyck Prefix Next Step Feasibility Tool Compute dyck prefix next step feasibility from explicit bounded inputs.
- Factorial Calculator Calculate n factorial for a nonnegative integer whose result is safely representable.
- Factorial Practice Create reproducible practice for Factorial Calculator from supplied scenarios, returning validated questions and computed answers under the source tool's assumptions and limits.
- Falling Factorial Last Factor Index Tool Compute falling factorial last factor index from explicit bounded inputs.
- Fibonacci Tiling Boundary Extension Tool Calculate fibonacci tiling boundary extension from explicit caller-supplied inputs.
- Fixed Weight Binary Word Rank Rank a fixed-weight binary word lexicographically by its zero-position combination, using exact zero-based integer text.
- Hypergraph Uniform Edge Incidence Count Tool Calculate hypergraph uniform edge incidence count from explicit caller-supplied inputs.
- Inclusion Exclusion Single Subset Sign Tool Compute inclusion exclusion single subset sign from explicit bounded inputs.
- Integer Composition Zero Part Insertion Tool Compute integer composition zero part insertion from explicit bounded inputs.
- Involution Single Fixed Point Budget Tool Compute involution single fixed point budget from explicit bounded inputs.
- Labeled Unlabeled Orbit Quotient Tool Divide an exact labeled-object count by a supplied common orbit size, requiring exact divisibility. Applicability of the common-size assumption is the caller's responsibility.
- Lattice Word Symbol Budget Tool Compute lattice word symbol budget from explicit bounded inputs.
- Matching Unpaired Vertex Budget Tool Calculate matching unpaired vertex budget from explicit caller-supplied inputs.
- Mixed Radix Digit Decoder Decode a zero-based exact integer index into mixed-radix digits most-significant lane first; out-of-range indexes do not wrap.
- Mixed Radix Digit Encoder Encode bounded mixed-radix digits most-significant lane first into exact integer text; every radix is positive and every digit must fit.
- Mixed Radix Next Digits Advance a mixed-radix tuple forward by one in most-significant-first lexicographic order, returning null rather than wrapping at boundaries.
- Mixed Radix Previous Digits Advance a mixed-radix tuple backward by one in most-significant-first lexicographic order, returning null rather than wrapping at boundaries.
- Mixed Radix Total Size Return exact mixed-radix product size as decimal integer text; an empty digit tuple has one possible state.
- Multinomial Category Merge Factor Tool Compute multinomial category merge factor from explicit bounded inputs.
- Multiset Swap Orbit Size Tool Count the orbit of one explicit pair under a single swap, including equal symbols and identical-position swaps.
- Multiset Word Adjacent Equality Count Tool Calculate multiset word adjacent equality count from explicit caller-supplied inputs.
- Necklace Orbit Size Tool Count distinct cyclic rotations of an explicitly supplied nonempty necklace; reflections are distinct.
- Necklace Period Divisor Flag Tool Necessary divisor condition for a cyclic word period; repetitions is the integer quotient.
- Necklace Rotation Fixed Position Count Tool Calculate necklace rotation fixed position count from explicit caller-supplied inputs.
- Next Combination Indexes Return the next lexicographic fixed-size index combination without wrapping; null marks the order boundary.
- Next Fixed Weight Binary Word Return the next lexicographic binary word with the same length and one-count, without wraparound; null marks the last fixed-weight word.
- Next Weak Composition Vector Return the next lexicographically increasing weak composition of the same total and length, preserving nonnegative integer parts; null marks the final vector.
- Orbit Size Divisibility Remainder Test explicit orbit-size divisibility of a finite group order without establishing that a group action or orbit exists.
- Partition Conjugate Single Column Height Tool Compute partition conjugate single column height from explicit bounded inputs.
- Permutation Adjacent Swap Inversion Delta Tool Compute permutation adjacent swap inversion delta from explicit bounded inputs.
- Permutation Count Calculator Calculate the number of ordered ways to choose k items from n items.
- Permutation Cycle Split Sign Tool Relative permutation sign from a supplied change in cycle count on the same underlying elements.
- Permutation Fixed Point Free Budget Tool Compute permutation fixed point free budget from explicit bounded inputs.
- Permutation Inversion Parity Compose Compose two explicitly supplied permutation parities using multiplication of signs, represented by Boolean oddness.
- Permutation Prefix Forced Fixed Count Tool Count already forced fixed positions in a supplied permutation prefix, without counting its possible completions.
- Permutation Transposition Count Parity Tool Compute permutation transposition count parity from explicit bounded inputs.
- Plane Tree Edge Vertex Balance Tool Check the necessary edge/vertex count identity for a nonempty tree; counts alone do not prove connectivity.
- Previous Combination Indexes Return the previous lexicographic fixed-size index combination without wrapping; null marks the order boundary.
- Set Partition Block Insertion Weight Tool Compute set partition block insertion weight from explicit bounded inputs.
- Subset Bitmask Integer Decoder Decode an exact bounded subset mask into ascending least-significant-bit indexes; masks with bits outside the explicit universe are rejected.
- Subset Bitmask Integer Encoder Encode a bounded unique index subset into an exact nonnegative integer bitmask, with index zero as the least significant bit.
- Subset Colex Leading Rank Tool Return the leading colex combinatorial-number-system term choose(maxElement,subsetSize) for a zero-based subset.
- Subset Complement Cardinality Shift Tool Calculate subset complement cardinality shift from explicit caller-supplied inputs.
- Subset Complement Indexes Complement a unique explicit index subset against a bounded 0..size-1 universe, returning ascending omitted indexes.
- Subset Fixed Intersection Budget Tool Calculate subset fixed intersection budget from explicit caller-supplied inputs.
- Subset Mask Cardinality Parity Tool Compute subset mask cardinality parity from explicit bounded inputs.
- Weak Composition Stars Bars Length Tool Calculate weak composition stars bars length from explicit caller-supplied inputs.
- Word Prefix Symbol Deficit Tool Compare a supplied prefix’s exact symbol counts with target multiplicities, preserving overfill and unknown tokens.